DOI: https://doi.org/10.26089/NumMet.v18r214

On solvability of a nonlinear system of equations for the Fourier-Chebyshev coefficients in the problem of solving ordinary differential equations using Chebyshev series

Authors

  • O.B. Arushanyan
  • S.F. Zaletkin

Keywords:

ordinary differential equations
approximate analytical methods
numerical methods
orthogonal expansions
shifted Chebyshev series
Markov’s quadrature formulas

Abstract

A solvability theorem for a nonlinear system of equations with respect to approximate values of Fourier-Chebyshev coefficients is formulated and proved. This theorem is a theoretical substantiation for the previously proposed numerical-analytical method of solving ordinary differential equations using Chebyshev series.

New computational technologies

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Published

2017-04-28

Issue

Vol. 18 (2017): Issue 2.

Section

Section 1. Numerical methods and applications

Author Biographies

O.B. Arushanyan

Lomonosov Moscow State University
• Head of Laboratory

S.F. Zaletkin

Lomonosov Moscow State University
• Senior Researcher

References

  1. O. B. Arushanyan and S. F. Zaletkin, “Approximate Solution of the Cauchy Problem for Ordinary Differential Equations by the Method of Chebyshev Series,” Vychisl. Metody Programm. 17, 121-131 (2016).
  2. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Calculation of Expansion Coefficients of Series in Chebyshev Polynomials for a Solution to a Cauchy Problem,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 5, 24-30 (2012) [Moscow Univ. Math. Bull. 67 (5-6), 211-216 (2012)].
  3. I. S. Berezin and N. P. Zhidkov, Computing Methods (Fizmatgiz, Moscow, 1962; Pergamon, Oxford, 1965).
  4. E. Hairer, S. P. Nörsett, and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems (Springer, Berlin, 1987; Mir, Moscow, 1990).
  5. R. England, “Error Estimates for Runge-Kutta Type Solutions to Systems of Ordinary Differential Equations,” Comput. J. 12 (2), 166-170 (1969).
  6. C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice Hall, Englewood Cliffs, 1971).
  7. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Application of Chebyshev Series to Integration of Ordinary Differential Equations with Rapidly Growing Solutions,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 5, 57-60 (2015) [Moscow Univ. Math. Bull. 70 (5), 237-240 (2015)].
  8. O. B. Arushanyan, N. I. Volchenskova, and S. F. Zaletkin, “Approximate Solution of Ordinary Differential Equations Using Chebyshev Series,” Sib. Elektron. Mat. Izv. 7, 122-131 (2010).

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